2017/11/02 by Xin Ma, Ma, Xin
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Operator Algebras (math.OA) #math.DS #math.OA
paper · pdf · doi:10.48550/arxiv.1711.00886
17 pages. Major revision. Enhancement of the reader experience by simplifying the proof of Lemma 3.1. Add a discussion of Definition 2.4 and a proof of Proposition 3.8. Update the reference list. To appear in JFA
openalex publication_date 2017/11/02 · openalex created_date 2017/11/10 · arxiv created 2018/06/28 · arxiv updated 2018/06/29 · openalex updated_date 2026/07/28
Let α: G\curvearrowright X be a minimal free continuous action of an infinite countable amenable group on an infinite compact metrizable space. In this paper, under the hypothesis that the invariant ergodic probability Borel measure space EG(X) is compact and zero-dimensional, we show that the action α has the small boundary property. This partially answers an open problem in dynamical systems that asks whether a minimal free action of an amenable group has the small boundary property if its space MG(X) of invariant Borel probability measures forms a Bauer simplex. In addition, under the same hypothesis, we show that dynamical comparison implies almost finiteness, which was shown by Kerr to imply that the crossed product is Z-stable. Finally, we discuss some rank properties and provide two classifiability results for crossed products, one of which is based on the work of Elliott and Niu.